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- Dictionaryquotient/ˈkwəʊʃnt/
noun
- 1. a result obtained by dividing one quantity by another.
- 2. a degree or amount of a specified quality or characteristic: "the increase in Washington's cynicism quotient"
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Jul 6, 2016 · You DO need an equivalence relation to build a quotient set, which is why the notation is S/~, which is read as "the quotient set of the set S under the equivalence relation ~." At the risk of over-simplifying it, you could say that the quotient set under a particular equivalence relation is the same as the original set, but in partitions rather than all together.
19. That operation on cosets is well-defined if and only if H is a normal subgroup. If H is just a subgroup, what you call "left quotient group" has the more standard name "set with a left group action". More precisely, the coset spaces G/H describe essentially all the examples of sets with transitive left G-actions.
Jul 18, 2018 · The axiom of choice is not needed here (assuming you can define your function $\psi:X\to Y$ without it and the resulting function $\phi$ is well-defined).
May 4, 2023 · Canonical definition of quotient bundle. For a vector bundle π: E → M π: E → M and a subbundle π′: E′ → M π ′: E ′ → M of E E where the ranks are r,r′ r, r ′, we can define the quotient bundle E/E′ → M E / E ′ → M. The usual construction is to find an open cover where there is a local frame σ1, ⋯,σr σ 1, ⋯ ...
Sep 11, 2011 · The "quotient topology" on X/R X / R is defined by saying " U ⊂ X/R U ⊂ X / R is open iff π−1(U) π − 1 (U) is open in X X." (the quotient topology itself is a special case of two things: 1) the weak topology induced by a famaily of maps too/from your set, and 2) a pushout diagram. You should check these things out, they're cool.)
Nov 7, 2016 · Multiplying on the right by (h′)−1 (h ′) − 1 and renaming h′′(h′)−1 =h1 h ″ (h ′) − 1 = h 1 gives that a′ = ah1 a ′ = a h 1. – davidlowryduda ♦. Commented Nov 6, 2016 at 21:10. Note that a′ a ′ is an element of a′H a ′ H, since H H contains the identity element. Therefore, if a′H = aH a ′ H = a H then a ...
Oct 12, 2016 · Quotient ring multiplication is saying that if you multiply any two elements of I + a I + a and I + b I + b, you get an element of I + ab I + a b, which is why the quotient map is well-defined and a homomorphism. It need not be possible to write every element of I + ab I + a b as the product of two elements in I + a I + a and I + b I + b, in ...
Jul 16, 2020 · The case of a norm will follow in part from this. Suppose that d is translation invariant metric on linear topological space X compatible with the linear topology τ. Let M be a closed linear subspace in X, π the quotient map, and τM the topology on X / M induced by π. Define ρ(π(x), π(y)): = inf {d(x − y, z): x ∈ M} = d(x − y, M ...
Jun 6, 2022 · Doubt on the notation of quotient sets and quotient vector spaces 6 For a normed space, is a vector subspace of it with the restriction norm also a normed space?
Defining a norm in the quotient space. E. /. M. Let (E, ‖ ⋅ ‖) a normed space and consider M ⊆ E a closed vectorial subspace. Consider in E the equivalence relation x ≡ y x − y ∈ M, and let E / M the quotient set. The equivalence class of x is the set x + M = {x + m | m ∈ M}.
